To determine if the function represented by the given table is linear or not, we need to check if the rate of change between the [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex] changes is constant.
Let's first list the given coordinate points:
[tex]\[
(6, 4), (7, 2), (8, 0), (9, -2)
\][/tex]
### Step 1: Calculate the Differences in [tex]\( x \)[/tex]-Values
The differences in [tex]\( x \)[/tex]-values (denoted as [tex]\( \Delta x \)[/tex]) are calculated between consecutive points:
[tex]\[
\Delta x_1 = 7 - 6 = 1, \quad \Delta x_2 = 8 - 7 = 1, \quad \Delta x_3 = 9 - 8 = 1
\][/tex]
So, the differences in [tex]\( x \)[/tex]-values are:
[tex]\[
\Delta x = [1, 1, 1]
\][/tex]
### Step 2: Calculate the Differences in [tex]\( y \)[/tex]-Values
Similarly, the differences in [tex]\( y \)[/tex]-values (denoted as [tex]\( \Delta y \)[/tex]) are calculated between consecutive points:
[tex]\[
\Delta y_1 = 2 - 4 = -2, \quad \Delta y_2 = 0 - 2 = -2, \quad \Delta y_3 = -2 - 0 = -2
\][/tex]
So, the differences in [tex]\( y \)[/tex]-values are:
[tex]\[
\Delta y = [-2, -2, -2]
\][/tex]
### Step 3: Calculate the Rate of Change
The rate of change (also known as the slope) between each pair of points is found by dividing the differences in [tex]\( Y \)[/tex]-values by the differences in [tex]\( X \)[/tex]-values:
[tex]\[
\text{Rate of change}_1 = \frac{\Delta y_1}{\Delta x_1} = \frac{-2}{1} = -2.0
\][/tex]
[tex]\[
\text{Rate of change}_2 = \frac{\Delta y_2}{\Delta x_2} = \frac{-2}{1} = -2.0
\][/tex]
[tex]\[
\text{Rate of change}_3 = \frac{\Delta y_3}{\Delta x_3} = \frac{-2}{1} = -2.0
\][/tex]
So, the rates of change are:
[tex]\[
\text{Rate of change} = [-2.0, -2.0, -2.0]
\][/tex]
### Step 4: Check if the Rate of Change is Constant
We observe that the rate of change is the same between every pair of points:
[tex]\[
-2.0 = -2.0 = -2.0
\][/tex]
Since the rate of change is constant, we can conclude that:
[tex]\[
\text{No, because it has a constant rate of change.}
\][/tex]
Thus, the function is linear.
